$x+y\ge 2\sqrt{x+y}$
$\to (\sqrt{x+y})^2-2\sqrt{x+y}\ge 0$
$\to \sqrt{x+y}(\sqrt{x+y}-2)\ge 0$
$\to \left[ \begin{array}{l}\sqrt{x+y}=0\\ \sqrt{x+y}-2\ge 0\end{array} \right.$ (do $\sqrt{x+y}\ge 0$)
$\to \left[ \begin{array}{l}x+y=0\\x+y\ge 4\end{array} \right.$